Existence and global stability of positive periodic solutions of a discrete delay competition system

Hai-Feng Huo,Wan-Tong Li

doi:10.1155/s0161171203210231

Abstract

The existence and the global stability of positive periodic solutions of a discrete competition model is studied. The model incorporates time delays and allows for a fluctuating environment. By means of some standard procedures of the topological degree method and the construction of a suitable Lyapunov function, sufficient conditions are obtained to ensure the existence and the global stability of positive periodic solutions of the above systems.

Highlights

There has been quite a lot of excellent work on the study of ecological competition systems
The classical two-species Lotka-Volterra autonomous competition model is described by the following systems of differential equations: u (t) = u(t) r1 − au(t) − bv(t), v (t) = v(t) r2 − cu(t) − dv(t), (1.1)
Motivated by all of the above, the principal aim of this paper is to study the global existence of periodic solutions of the following discrete delay competition system: u(k + 1) = u(k) exp r1(k) − a(k)u k − n1 − b(k)v k − n2, v(k + 1) = v(k) exp r2(k) − c(k)u k − l1 − d(k)v k − l2, (1.3)

Summary

Introduction

There has been quite a lot of excellent work on the study of ecological competition systems (see [3, 4, 11, 16] and the references therein). Motivated by all of the above, the principal aim of this paper is to study the global existence of periodic solutions of the following discrete delay competition system: u(k + 1) = u(k) exp r1(k) − a(k)u k − n1 − b(k)v k − n2 , v(k + 1) = v(k) exp r2(k) − c(k)u k − l1 − d(k)v k − l2 ,. Few papers investigate the global stability of positive periodic solutions of these models. We use the Mawhin’s continuation theorem of the coincidence degree theory to investigate the existence of at least one positive periodic solution of system (1.3).

Now we take

We define the function V by